To illustrate the derived result eq:prop1 and provide numerical evidence of the conjecture eq:conj, we plot in fig:p1The simulations performed in fig:p1–3 make use of the Mathematica codes provided by Santosh Kumar based on the log-gas approach as discussed in[1]}. the exact variance of von Neumann entropy as compared with the simulations. We consider the scenarios of equal and unequal subsystem dimensions, where the claimed results in solid lines match well with the corresponding simulations as represented by the scatters. We also observe that the slope of the curves approaches to zero when the dimension of the system becomes larger. This phenomenon can be explained by the asymptotic formula eq:valim of the variance, i.e., the variance is dominated by the ratios of subsystem dimensions for large-dimensional systems.
